Comparing Linear, Polynomial, and Exponential Growth

Suppose you graph all three on the same axes, and ask the question, “Which function grows the fastest for large values of
x
?”

In this graph, it appears that the answer is
y
=
x
4
. It is greater than
y
=
2
x
for all values shown.

However, if we zoom out the
y
-axis quite a lot, we see that at
x
=
16
,
y
=
2
x
overtakes
y
=
x
4
, and after this point,
y
=
2
x
grows faster. (Note that with the axes scaled this way,
y
=
x
grows so slowly that it is indistinguishable from the
x
-axis.)

In fact, it can be shown that this is true for ANY exponential function and ANY polynomial function with positive growth. The exponential function will eventually outstrip the polynomial function.